Kurzweil-Henstock Integral in Riesz spaces - Bentham Science

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7 Improper Integral 84 7.1 Real valued case 84 7.2 Vector valued case 99 8 Choquet and Šipoš Integrals 111 8.1 Symmetric Integral 111 8.2 Asymmetric integral 126 8.3 Applications 127 9 (SL)-Integral 134 9.1 Main properties in the real and Riesz space context 134 9.2 Convergence theorems 158 10 Pettis-Type Approach 166 10.1 Banach space valued case 166 10.2 Riesz space valued case 172 11 Applications in Multivalued Logic 178 11.1 MV-algebras 178 11.2 Group-valued measures 181 11.3 Intuitionistic fuzzy sets 184 12 Applications in Probability Theory 187 12.1 Independence 187 12.2 Conditional probability 191 12.3 Probability theory on IF-events 194 13 Integration in Metric Semigroups 198 13.1 Elementary properties 198 13.2 Convergence theorems 208 References 213 Subject Index 224
FOREWORD In 1956 Jaroslav Kurzweil was involved in special phenomena occurring in ordinary differential equations with fast oscillating entries which have not been justified by theories and approaches known at those times. For explaining the observed results he constructed a tool, which reminded in some aspects strongly the way how the Perron integral using minor and major functions was constructed. The story ended by a success and the tool became an independent, self-contained object, the generalized Perron integral. Since the generalized Perron integral occurred to be very interesting and, due to the need of research, it was described via integral sums of Riemann type, Jaroslav Kurzweil described it in detail in his very first paper [163] on the topic and used it in a series of subsequent writings on ordinary differential equations. Only a restricted number of people knew at this times about the existence of a newly defined integral. There is no reason to be surprised by this fact, looking at the title of [163] nobody can expect deep interest of mathematicians involved in integration theory in this paper. In the same time Ralph Henstock worked on variational approaches to integrals, no existing connection to Kurzweil in those days. For the first time the possible relation is mentioned cautiously in Henstock’s booklet [153]. It was discovered early in the sixties that in the case of real valued functions both approaches (that of Henstock and of Kurzweil) are equivalent and, of course, the definition of the very general non-absolutely convergent integral based on Riemann-type integral sums came to the foreground. The distinctive individual life of an integration theory, the Kurzweil-Henstock integral, started in the second half of seventies. With all of its advantages and drawbacks coming to general awareness. One of the interesting points of integration theories is the problem when functions with values in general spaces have to be integrated. This problem is of interests especially in the case of infinite dimensional spaces equipped with some topology, the models are e. g. the Bochner, Dunford and Pettis integrals of Banach spacevalued functions based on the Lebesgue approach. The book of A. Boccuto, B. Riean and M. Vrábelová is oriented in this direction. The functions which are integrated have values in Riesz spaces in general. The combination of techniques used in Riesz spaces with the more or less algebraic approach which is in charge for integrals based on Riemann type integral sums makes the presentation interesting and inspirative for further research. Besides their own research the authors present also short insights into applications and less known theories. All this makes the value of the present book, which should reach the reader in an unorthodox form. I am sure we are facing an inspirative text, with many new information and maybe also a nice reference text in various fields of analysis. tefan Schwabik, Prague i
Page 2 and 3: Kurzweil-Henstock Integral in Riesz
Page 6 and 7: ii PREFACE Kurzweil and Henstock’
Page 8 and 9: iv KURZWEIL-HENSTOCK INTEGRAL IN RI
Page 10 and 11: 2 Kurzweil-Henstock Integral in Rie
Page 14 and 15: Elementary Theory of Riesz Spaces K
Page 16 and 17: Kurzweil-Henstock Integral with Val
Page 18 and 19: 52 Kurzweil-Henstock Integral in Ri
Page 20 and 21: 54 Kurzweil-Henstock Integral in Ri
Page 22 and 23: Kurzweil-Henstock Integral in Topol
Page 24 and 25: 6 Convergence Theorems Kurzweil-Hen
Page 26 and 27: Convergence Theorems Kurzweil-Henst
Page 28 and 29: Improper Integral Kurzweil-Henstock
Page 30 and 31: 8 Choquet and ipo Integrals Kurzwei
Page 32 and 33: Choquet and ipo Integrals Kurzweil-
Page 34 and 35: (SL)-Integral Kurzweil-Henstock Int
Page 36 and 37: 166 Kurzweil-Henstock Integral in R
Page 38 and 39: 168 Kurzweil-Henstock Integral in R
Page 40 and 41: Applications in Multivalued Logic K
Page 44 and 45: Applications in Probability Theory
Page 46 and 47: Integration in Metric Semigroups Ku
Page 50 and 51: References Kurzweil-Henstock Integr
Page 52 and 53: Index Kurzweil-Henstock Integral in

Kurzweil-Henstock Integral in Riesz spaces - Bentham Science

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